Mesh Sensitivity, Rate Effects & Numerical Robustness

What Is It?

Mesh sensitivity, strain-rate effects and numerical robustness are the factors that determine whether the results of an explicit analysis are reliable. Mesh sensitivity means the predicted response — particularly failure — changes with mesh refinement. Strain-rate effects mean the material behaviour depends on how fast it is deformed. Numerical robustness means the analysis produces consistent results despite small changes in inputs. Understanding and controlling these factors is essential for credible explicit analysis, particularly when failure is involved.

Why It Matters

An explicit analysis that includes material failure can produce different results depending on the mesh size, the strain-rate model and the numerical controls. A finer mesh may predict failure at a different load or in a different location. A different strain-rate model may predict different material behaviour. Changes in contact stiffness, hourglass control or artificial viscosity may change the deformation pattern. Without understanding and controlling these sensitivities, the engineer cannot determine whether the predicted result is a physical prediction or a numerical artefact.

Mesh Size and the Stable Time Step

Mesh size affects both the accuracy and the cost of explicit analysis. A finer mesh captures stress gradients and local deformation more accurately but has a smaller stable time step, increasing computation time. A coarser mesh is cheaper but may miss local features. The mesh should be fine enough to resolve the smallest feature of engineering interest — the contact zone, the stress concentration, the failure region — but not so fine that the computation becomes impractical. The mesh size should be determined by the physics, not by the available computational resources.

Damage Localisation and Mesh Dependency

When a material model includes strain softening — stiffness decreasing with damage — the failure tends to localise into a narrow band. The width of this band is controlled by the mesh size: a finer mesh produces a narrower localisation band. This means the energy dissipated during failure depends on the mesh — a finer mesh dissipates less energy in the failure zone because the zone is narrower. This is mesh dependency of failure — the predicted failure load and failure energy change with mesh refinement. Without regularisation, the failure prediction does not converge to a unique value as the mesh is refined.

In a failure model, mesh refinement can change the failure energy unless the formulation is regularised appropriately. A mesh-dependent failure prediction is not a physical prediction — it is a numerical artefact that changes with mesh size.

Regularisation and Characteristic Length

Regularisation is the technique used to make failure predictions mesh-independent. The most common approach is to scale the damage evolution law by the element characteristic length. The energy dissipated per unit area of fracture (the fracture energy Gc) is kept constant, and the stress-strain response in the damage zone is adjusted based on the element size. This ensures that the total energy dissipated in the failure zone is the same regardless of mesh size. The characteristic length is a property of the element — typically related to the element dimension or volume. Regularised damage models converge to a unique failure prediction as the mesh is refined.

Regularised damage evolution (concept):

Damage energy per element  =  G_c · A_fracture

where:
G_c        =  fracture energy (material property, J/m²)
A_fracture =  fracture area associated with element

Element strain at failure scales with element size:
  ε_f  ∝  G_c / (σ · L_c)

where L_c = characteristic element length

→ Finer mesh: smaller ε_f, but same total fracture energy

Strain-Rate Effects

Many materials — particularly metals — exhibit rate-dependent behaviour at the high strain rates encountered in impact. The yield strength and flow stress are higher at higher strain rates. This rate dependence is important for impact analysis — the material may appear stronger during the rapid loading of an impact than it would under quasi-static loading. Rate-dependent material models require rate-dependent material data — high-rate test data obtained from split-Hopkinson-bar tests or other dynamic test methods. Using quasi-static material data for a high-rate impact analysis may under-predict the material strength and over-predict the deformation.

High-Rate Test Data Requirements

If the analysis includes strain-rate effects, the material data must cover the strain rates expected in the analysis. Using a rate-dependent model with only quasi-static data is meaningless — the rate dependence is extrapolated without basis. High-rate test data is difficult and expensive to obtain — it requires specialised equipment and careful calibration. If high-rate data is not available, the engineer must either obtain it, use a rate-independent model (acknowledging the limitation), or use conservative assumptions. The basis for the rate-dependent data should be documented.

COMMON MISTAKE: Using a rate-dependent material model without rate-dependent material data. The rate dependence is then an extrapolation with no physical basis. If rate effects are included, the material data must cover the expected strain-rate range.

Artificial Viscosity

Artificial viscosity is a numerical technique used to stabilise shock waves in explicit analysis. When a shock wave (a sharp discontinuity in stress and velocity) propagates through the mesh, the discontinuity can produce oscillations — numerical noise around the shock front. Artificial viscosity adds a viscous pressure that smooths the shock over a few elements, eliminating the oscillations. The amount of artificial viscosity is a numerical parameter — too little allows oscillations; too much smears the shock excessively. Artificial viscosity should be used where shock waves are present (high-velocity impact, blast) but should be minimised for low-velocity problems where shocks are not significant.

Sensitivity Studies

For explicit analyses involving failure, contact and high-rate loading, sensitivity studies should be performed to assess the robustness of the result. The key parameters to vary are friction coefficient, mesh density, material data (particularly failure strain or damage parameters), mass scaling amount and contact stiffness. If the result is insensitive to these parameters, confidence is high. If the result changes significantly, the parameters must be better defined or the result must be reported with its sensitivity range.

• Mesh sensitivity — refine mesh and check if failure prediction changes
• Friction sensitivity — vary friction coefficient within plausible range
• Material data sensitivity — vary failure parameters within data scatter
• Rate effect sensitivity — compare rate-dependent and rate-independent results
• Mass scaling sensitivity — check if mass scaling changes the dynamic response

Key Takeaways

• Mesh sensitivity in failure models is a fundamental issue — regularisation is needed for mesh-independent predictions
• Strain-rate effects require rate-dependent material data covering the expected strain-rate range
• Artificial viscosity stabilises shock waves but should be minimised for low-velocity problems
• Hourglass modes and element formulation affect the numerical robustness of the result
• Sensitivity studies on mesh, friction, material data and numerical controls are essential for credible results